"""
====================================
7.01 Inverse Laplace Operator
====================================
We reproduce here the figure 7.1 and 7.2 of the book.
Utilitary functions can be found next to this file. Here, we only define codpy-related functions.
"""

#########################################################################
# Necessary Imports
# ------------------------
import os 
import sys 

import matplotlib.pyplot as plt 
from codpy import core

try:
    CURRENT_DIR = os.path.dirname(os.path.abspath(__file__))
except NameError:
    CURRENT_DIR = os.getcwd()
data_path = os.path.join(CURRENT_DIR, "data")
PARENT_DIR = os.path.abspath(os.path.join(CURRENT_DIR, ".."))
sys.path.insert(0, PARENT_DIR)

from utils.ch7.ch7_utils import PDE1, PDE2

#########################################################################
# Problem statement
# ------------------------
# We solve the following Poisson equation with Dirichlet boundary conditions: 
# $$\Delta u  = f, \quad \text{ supp } u  \subset \Omega, \quad u_{\partial \Omega}=0$$
# where $f$ is sufficient regular and $\Omega$ is a sufficient regular domain.
# To do so we select a mesh, and a kernel, which approximates this equation by approximating the solution as a function $u \in \mathcal{H}_k$
# This leads to the equation $(\Delta_k u)(X) = f(X)$, $\Delta_k$ being the approximation of the Laplace-Beltrami operator defined in the book.
# A solution to this equation is computed as $u = (\Delta_k)^{-1} f$.

#########################################################################
# Regular mesh
# ------------------------
# This figure displays a regular mesh for the domain $\Omega = [0,1]^2$, where $f$ is plotted in the left-hand side, and the solution $u$ in the right-hand side.

PDE1() 
plt.show()

#########################################################################
# Irregular mesh
# ------------------------
# This figure computes a Poisson equation on an unstructured mesh generated by a bimodal Gaussian random variable, with $f$ plotted on the left and the solution $u$ on the right.

PDE2() 
plt.show()